Multi-peak bound states for nonlinear Schrödinger equations

A NNALES DE L ’I. H. P., SECTION C

M ANUEL D EL P INO

P ^ATRICIO L. F ^ELMER

Multi-peak bound states for nonlinear Schrödinger equations

Annales de l’I. H. P., section C, tome 15, n

^o

2 (1998), p. 127-149

<http://www.numdam.org/item?id=AIHPC_1998__15_2_127_0>

© Gauthier-Villars, 1998, tous droits réservés.

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RESUME. - Dans cet

article,

^on considere 1’etude des solutions de ondes permanentes ^d’une

equation

^de

Schrodinger

nonlinéaire. Ce

problème

^se

reduit a la recherche de solutions non

negatives

^de

avec une

energie

finie. Ici c est un

parametre petit,

^{SZ est un} domaine lisse

qui peut

^{etre non}

borné, fest

^{une fonction}

superlineaire appropriee

^{et V}

est un

potentiel positif

borne hors de zero.

L’objectif

^{de cet} article consiste a obtenir des solutions a

pics multiples

dans le cas de

puits multiples.

^{Nous trouvons} des solutions

qui

^montrent

une concentration pour ^tout ensemble choisi fini de minima locaux du

potentiel, qui

peuvent ^etre

degeneres.

La demonstration se base ^sur des arguments

variationnels,

^{ou une methode} de

penalisation

^est

developpee

pour identifier les solutions cherchees.

©

Elsevier,

^Paris

0. INTRODUCTION The nonlinear

Schrodinger equation

ⁱⁿ

has been

object

of extensive research in recent years. In this paper ^we consider the

study

^of

standing

^{waves of}

equation (o.1 ), namely

^of

special

solutions of the form

~~(~;. t)

^{= where}

v(x)

^{> 0. It is}

easily

checked that

a 03C8

of this form satisfies

equation (0.1)

^{if and}

only

^if

the function

2l ( ~~ )

solves the

elliptic equation

The

problem

^{we will}

study

in this paper is that of existence of

positive

solutions with finite _energy to this

equation

^{when is}

strictly positive.

away from _zero, and the

potential

W exhibits

multiple

^wells,

namely

several,

possibly degenerate

local minima. 11, will be

regarded

^{as a small} parameter. ^After

absorption

^{of the} parameters

by scaling,

^the

problem

under consideration may be rewritten ^as

cle I ’ht.stittrt Analyse ^{non linéaire}

where V ^> a > 0 in N > 1. In

[5],

Floer and Weinstein consider the case N ⁼ 1 and _p ⁼ 3. For a

given nondegenerate

^critical

point

^{of the}

potential ^V,

^assumed

globally ^bounded, they

^{construct a}

positive

^solution

r._ ^to

(0.3), provided

^{that c is}

sufficiently

small. This solution concentrates

around the critical

point

^{as ~ -}

^0,

^{in the sense that its}

shape

^{is a}

sharp peak

^{near that}

point,

while it almost vanishes

everywhere

^else.

Their

method,

^{based on an}

interesting Lyapunov-Schmidt

^finite

dimensional

reduction,

^{was extended}

by

^{Oh in}

[8], [9]

^{to conclude a}

similar result in

higher dimensions, provided

^{that 1 p}

~_±2.

Oh restricts himself to

potentials

with "mild oscillation" at

infinity,

^name-

ly belonging

^{to a Kato class. In case that} V is bounded this restriction is not necessary ^as noticed

by Wang

ⁱⁿ

[14]. Wang

also observes that if V is nonconstant and

nondecreasing

^{in one}

direction,

^then

equation (0.3)

^has

no solutions which tend to zero at

infinity.

The method in

[5]

^and

[8]

^{seems to}

rely

ⁱⁿ essential way ^{on the}

nondegeneracy

of the critical

points.

^In

[ 11 ],

Rabinowitz lifted

partially

^this

requirement introducing

^a

global

variational

technique

^{to find a solution}

with "minimal

energy"

for all small ~, when 1 p

N±~

^and

Rabinowitz’s

approach actually

^{covers a} broader class of nonlinearities and the smallness of ~ is not

required

^{in case} that the limit in the left of

(0.4)

is -f- ~c. In

[14], Wang

established that this least energy

(mountain pass)

solution indeed concentrates around a

global

minimum of ‘T in the

special

case of

equation (0.3),

^{as c - 0.}

In

[3],

^the authors succeeded in

proving

the local

analogue

^of

Wang’s

result. It is shown that if for an open, bounded ^{set one has}

then

(0.3)

possesses solutions ~c~ with

just

^{one local}

^maximum,

^which

concentrates around a minimum of V in A. This local minimum may exhibit

arbitrary degeneracy.

Concerning

^{solutions with}

multiple

concentration

points,

ⁱⁿ

[10]

^Oh

applies

^the

approach developed

ⁱⁿ

[5]

^and

[8]

to construct a

family

^of

solutions _~u= with

peaks concentrating

^around any

prescribed

^{finite set of}

nondegenerate

^critical

points

^{of V when N = 1, and indicates how to}

proceed

ⁱⁿ

higher

dimensions where 1 p

~~.

It is the purpose of this article to obtain

multi-peak

^{solutions of}

(0.3)

ⁱⁿ

the

"multiple

^well

case", exhibiting

concentration at any

prescribed

^{set of}

Vol. 15. n ~ 2-1998.

possibly degenerate

local minima of the

potential, namely

^{on K}

disjoint

sets A where

(0.5)

^holds.

We will

actually

^{consider a more}

general

semilinear

elliptic problem

ⁱⁿ

a smooth domain

SZ, possibly unbounded,

of the form

The

potential

V will be assumed

throughout

this paper

locally

^Holder

continuous and bounded below _away from zero, say

We will also assume

that f : ^{f~+ -~ l~}

is of class

Cl

and satisfies the

following

conditions.

(f4)

^The

limiting equation

possesses ^a

unique ^solution,

up ^to

translation,

for _any

given

^{b > 0.}

Our main result for

equation (0.6)

^{is the}

following.

THEOREM 0.1. - Assume that there are bounded domains

i, mutually disjoint, compactly

^{contained in}

^SZ,

^{i =}

1.... , K,

^{such that}

Then there is an ~0 ^> 0 such

that for

every 0 ^~ ~0 ^a

positive

^solution

u~ E

H10 (03A9)

^to

problem (0.6)

^{exists. Moreover,} u~ possesses

exactly

^{K local}

maxima x~,i with x~,i in

Ai.

^{We also have that ~ infAi}

V,

^and

for

^{all x}

~ 03A9B~j~ij, where

^{cx are certain}

positive

^constants.

Amules de 1’Irrstitrrt Henri Poincaré - Analyse ^{non lindaire}

We observe that no restriction ^on the

global

behavior of V is

required

other than

(0.7).

^In

particular,

^{V is not}

required

^to be bounded or to

belong

to a Kato class.

On the other

hand,

^the

hypotheses on f

^are milder than those

required

in

[10].

^In

particular, hypothesis (f4)

is satisfied

by

^a

large

^{class of}

nonlinearities

f including

^{p > 1. See the work in}

Kwong

^and

Zhang [6],

and Chen and Lin

[ 1 ]. Moreover,

^this

assumption

^{can be} further relaxed to

(f4’)

^The mountain pass value in of the _energy functional associated to

(0.9)

is the smallest nontrivial critical value and it is isolated.

See the remark at the end of

§2.

It is

interesting

^to observe the

relationship

between the result of Theorem 0.1 and the work

by

Coti-Zelati and Rabinowitz

[2]

^{on multi-}

bump

solutions in

spatially-periodic problems.

^{For the case of}

equation (0.3),

¹ p

l~r~~’ ,

^{and V}

^periodic

^on each of its

variables,

^{it follows} from their results that for a

fixed,

^not

necessarily

^{small ~,} solutions in with

exactly

^K

bumps

exist for each

integer K, provided

^that

the associated energy functional satisfies the so-called

(*) nondegeneracy

condition. This

assumption

^states that all critical

points

^{at energy levels in}

~c.

^{c +}

8)

^are

^isolated,

^{where c} denotes the mountain pass minimax value of the associated

functional,

^{and 8 > 0.} These solutions have energy level close to Kc. On the other

hand,

Theorem 0.1 is

applicable

to construct a

K-bump

solution in this situation when 6- is

small, just assuming

^{that V} possesses

one local minimum. We do not know whether

(*)

^{holds in this case.}

The

proof

of Theorem 0.1 is

variational,

^{and uses} ideas in the

spirit

^of

those in our

previous

^work

^[3],

^{where a}

penalization

^method

enabling

^the

identification of local mountain passes ^was

developed. Roughly speaking,

the main

argument

^there consists of

defining

^{a suitable} modification of the

nonlinearity

for which the mountain pass theorem is

directly applicable

^to

the associated _energy.

Then, taking advantage

^{of the}

energy-minimality

^of

the mountain _pass

solution,

^one

finally

^{shows that it becomes a solution to}

the

original problem

^{with the} desired characteristics when E is

sufficiently

small.

Our current framework is more

delicate,

^{since the solutions we look for}

are at

higher

energy levels.

They

^{are not}

just rough

^mountain passes, ^so that

energy-minimality

^{is lost. We are able to overcome this}

difficulty by adding

^{a new}

penalization

^{term. In}

fact,

^{we introduce a} modification of the

nonlinearity

^{similar to that in}

[3]

^{in order to} prevent ^the occurrence of concentration outside the open ^sets and then a new term which

provides

a "balance" among the

energies

^{inside the distinct in order to obtain}

exactly K-bumps.

The solution is

captured

^{as a}

simple

^minimax

quantity

Vol. 15. n ~ 2-!998.

on a class of K-dimensional maps, and

eventually

^{shown to be a solution}

to the

original equation

^{with the}

appropriate

^features.

Finally,

^we would like to mention that the construction of solutions of

(0.3)

^{with an} infinite number of

bumps (hence

^not with finite

energy)

has been

recently

^{carried out}

by

^Thandi

[13]

^{in the}

nondegenerate

^case.

Infinite-bump

solutions in the framework of

[2]

^{were found}

by Spradlin

in

[12].

The

organization

^{of this} paper is ^as follows:

In §1

^{I we} define the modification of the functional needed for the

proof

of Theorem

0.1,

^and prove ^some

preliminary

^results.

§2

is devoted to the

proof

of Theorem 0.1.

1. PRELIMINARIES

This section is devoted to the definition and

preliminary study

^{of the}

penalized

functional.

We introduce an

appropriate penalization

^{so that} the concentration outside the sets

11~

is avoided and an

adequate

balance in the concentration is achieved. Then we prove that the

penalized

functional satisfies the Palais Smale condition

(P.S.),

^{and we set} up ^the minimax scheme in order to obtain critical

points

of it. We

provide

^{next some estimates on} the critical

points.

In the framework of Theorem

0.1,

associated to

equation (0.6)

^{we have}

the

"energy"

^functional

which is well defined for u G

H,

^where

H becomes a Hilbert _space,

continuously

^{embedded in when}

endowed with the inner

product

whose associated norm we denote

In the definitions

above,

and in what follows we assume that

f ,

^{V and Q}

satisfy

^the

hypotheses

^{of Theorem 0.1,} and that the

function f

^{is extended}

Annales de l’Institut Henri Poincaré - Analyse ^{non linéaire}

as 0 on the

negative

axis. Under those

assumptions

^{it is standard to check}

that the nontrivial critical

points

^of

~~ correspond exactly

^{to the}

positive

classical solutions ⁱⁿ of

equation (0.6).

Next we define the first modification of our functional. Let 0: be as in

(0.7),

^{and let us choose a > 0 so that}

This choice of a can be made since

(fl)

^and

(f3)

hold. Let us set

and define

where A ⁼

~Ki=1i,

with the bounded domains

Ai

^{as in the}

assumptions

of Theorem

0.1,

^and _x,1

denoting

^its characteristic function. It is easy ^to check that

(fl)-(f3) implies that g

defined in this way is ^a

Caratheodory

function and it satisfies the

following

Here we have denoted

G(x, ~) _ ~ fo ^T)dT.

Now we define the modified functional

J~ :

^{j~f -~ R as}

The functional

J~

^{is of class}

^Cl

^{in H and} its critical

points

^{are the}

positive

solutions of the

equation

Next we introduce the second modification of the functional. For this purpose ^we assume, without loss of

generality,

^{that the sets}

A~

have smooth

Vo). 15. rr 2-1998.

boundary.

^We define the

numbers b.l

^{= E and we let}

8 > 0 be so that

We will need the

’limiting

functional’

I b :

-~ f~ defined as

whose mountain pass critical value will be denoted

by c(b) .

^Critical

points

^of

Ib

are the solutions of

(0.9).

^{We define} Ci ⁼ and _(Ti ⁼

c(bi

⁺

8) -

^ci.

We assume that

This can be achieved

by making Ai

^{and 8} smaller if necessary. ^It will be convenient to consider

mutually disjoint

open ^sets

Ai compactly containing Ai,

^{for all i =}

1,...,

^{K. Then we define on H} the functional

and the

penalization

The constant M will be chosen later.

Finally

^the

penalized

^functional

E- : H ~ f~

^{is defined as}

The functionals

~~

^and

P

^{are of class}

^C1

^{and so is}

E~.

^{We show next}

that

E~

^has

good

compactness

properties,

^{that is}

E~

^satisfies the Palais Smale condition.

LEMMA 1.1. - Let

{un}

^{be a} sequence in H such that

E~(un)

^{is bounded}

and E’~(un) ^~ 0. Then un has a convergent

subsequence.

Annales de l’Institut Henri Poincaré - Analyse ^{non lindaire}

Proof -

^We first prove that the sequence is bounded in H.

Using property (g3)

^we

easily

^{see that}

from

where, using (g3) again,

^{we obtain}

In a similar fashion ^we find

We observe that

(1.13)

still holds if we

replace B by

any

number 8

E

(2, B).

Then,

for the

penalization

functional ^we have

From where there exists a constant C so that

We observe that the constant C is a

multiple

^of

Thus,

it follows from

(1.12), (1.15)

^{and the}

assumptions

^{on that}

this sequence ^is bounded as desired.

Let us choose a

subsequence,

still denoted

by weakly convergent

to u in H. This convergence is

actually

strong.

Indeed,

it suffices to show

that, given

q ^>

0,

^{there is an R > 0 such that}

where

BR

denotes the ball with center 0 and radius R. We may ^assume that R is chosen so that A c

Vol. 15, nv 2-1998.

Let r~~ be a cut-off function ^so that _1JR ⁼ 0 on ⁼ 1 on

S2 ~ ^BR.O

^{rlR 1}

and

^{Since is a} P.S. sequence,

we have that

where

on (1)

^{- 0 as n -~ oc .}

^Then,

i s bounded

We conclude that

from where

(1.16)

^{follows. D}

The

previous

^{lemma makes}

possible

^{to use} Critical Point

Theory

^to

find critical

points

of the functional

E~.

^We will formulate ^an

appropriate

minimax

problem

^for

E~.

We start

defining

^{a class of functions r over which we minimax. A}

continuous

function 03B3 : [0,1]K

~ H is in F if there are continuous

functions gi : [0, 1]

^~

^H,

^{u = 1, ... ,} K such that

(i)

^C

Ai

^{for all T} E

[0,1],

(ii) ~(T~, ..., Th ) _ ~~1 ~ gl (Tl,~

^{for all}

(Tl, ..., Th’~ ~ (~[~, l,Ii ~

(iii) g1 (0)

^{= 0 and}

~Ic (g~ (1))

⁰

(iv) E~ (’Y(t) ) ~ ‘~ (~ ~ c~ - ~r)

^{for all t E}

c~~0, l~ ~ ,

where 0

z

⁼

1, .... K ~

^{is a} fixed number.

We can define the minimax value associated to the class r as follows

In the

following

^{lemma we}

provide

^the

key

^{estimates on} the minimax value

C~.

^In

particular

^we show that F is nor empty.

LEMMA 1.2.

Annales de l’Institut Henri Poincaré - Analyse ^{non linéaire}

and in what

follows,

^{we denote}

by o(1)

^a

quantity approaching

^zero

-~ 0.

The

proof

^{of this lemma will}

require

^the

study

^{of an}

auxiliary

^Neumann

problem.

^{We may} consider the the functional

J1

^on

^Hl (11i ) .

^Let

d~

^{be the}

mountain pass value of

namely

where

F,

is the class of all continuous curves :

[0, 1]

^-~

H1 (11.~, )

^such

that

~yl (0)

^{= 0 and 0. Then we have}

LEMMA 1.3. - The mountain pass critical value

df ^of

^{the Neumann}

problem satisfies

For the sake of

continuity

^{in the} arguments, ^we

postpone

^the

proof

^of

this lemma to the

Appendix.

Proof of Lemma

1.2. - Since _ci is the mountain pass value for the

limiting

functional

given any 8

^{> 0} there exists a

path 03B3i : [0, 1]

^~

^H1 (RN)

such that

~y.z(0)

⁼

^{0, Ibi} (~y2(1))

^{0 and}

We assume from now on that 8

miry ~, ^Next, given ~

^>

^0,

^we

define the

path §.1 :

^:

[0,1]

^{-~ H as}

Here x.i

^E

Ai satisfies b.L

⁼

Y(xi), and ~i

^{is a C°° cut} off function with

compact

support ⁱⁿ

A,, taking

^{the value 1}

except

^{for a small}

neighborhood

of It is not hard to check that

with

o( 1 ) independent

^{of T E}

~0, 1].

Now we define the continuous function _{1’0 :}

~0. l~ ~’

^{-~ H as}

Vol. 15. nr 2-1998.

The function _fO

belongs

^{to F. In}

^fact, properties (i)-(ii)

^are

trivially

^true,

and

(iii)-(iv)

^are satisfied in view of

(1.19)

^and

(1.20),

^{when E is small}

enough. ^Thus,

the minimax value

C’l

^{is well defined as a real}

number,

^for all E > 0 small

enough.

We observe

that, by

the choice of 6 and from

(1.19)

^and

(1.20),

^the

penalization

^{term in}

jE~

^{vanishes on for all t G}

~0. l~ ~’ .

We also obtain from

(1.19)

^and

(1.20),

that for small E

so that

As 6 is

arbitrary,

^{we obtain the} upper estimate

C~ ~l~’T ( ~.~ 1 cL

⁺

^{o( 1 ) ) .}

We prove the lower estimate next. First ^we observe that

given

any r ^E F and _any curve

c(s) joining ~0~

^x

~0,

^with

~T ~

^x

~0, T~~’ -1,

^the

path

and then it follows from Lemma 1.3 that

We have an

inequality

^{of this form for every Ii =}

1,...,

^{K. Thus we can}

repeat

^the

argument given by

Coti-Zelati and Rabinowitz in

[2]

^{in the}

proof

of

Proposition

^{3.4 to}

^obtain,

^for every "( ^G

F,

^the existence of a

point

t E

~0 . T ~ ~’

^{such that}

From

here,

and from the form of

~T‘

^{outside the}

A;’s,

^{we have}

finishing

^the

proof.

^D

Annales de l’Institut Henri Poincaré - Analyse ^{non linéaire}

The functional

E~

satisfies the Palais Smale

condition,

the class r is not

empty,

^{and estimate}

(1.17) holds,

then there exits a critical

point

^{~c~ E H}

of

E~

^{such that}

E¿( UE)

⁼

^CEo

We define the local

weight

and then the function

The critical

point

^{~c~ is a} weak solution of the

equation

so that _~c~. satisfies

for _every set 0 ~ 03A9 not

intersecting ~(~Ki=1i).

We define the sets

~

⁼

{?/

^e

A~

⁼

(y

^{e ~ e}

AJ

and

= ~ ~

^{E E}

11.L ~ .

^We rescale the function _~c~ as

v~(y)

⁼

for y

^{E This}

function v~ belongs

^{to and}

then to and it satisfies in a weak sense the

equation

and over sets

0,

subsets of

5~~.

^not

intersecting

^{the function}

~ satisfies

Finally, proceeding

^{as in the first}

part

^{of the}

proof

^{of Lemma}

^1.1,

^{we obtain}

from the estimates on

C~ given

^{in Lemma 1.2 that}

and then for the

function v~

^we have the uniform

H1-estimate

Vol. 15, rr 2-1998.

2. PROOF OF THEOREM 0.1

In this section we will carry ^out the

proof

^{of Theorem 0.1.}

Using

^the

estimates obtained in the

previous ^section,

^we will show that if M in the definition of

E~

is chosen a

priori sufficiently large, then u~

^{will be a}

critical

point

^{of the}

original

functional

7~

whenever ~ is

sufficiently

^small.

Toward this

end,

^the

following

lemma constitutes ^a crucial

step.

^{It tells}

us in

particular

^{that if M was}

large

^{then the}

penalization

^term

P~ (u~ )

becomes zero for all

sufficiently

^{small ~.}

LEMMA 2.1. -

If in

^the

definition of E~

ⁱⁿ

(1.10)

^and

(l.ll ),

^{M > 0 was}

chosen

sufficiently large,

^then

For the

proof

^of this result some

preliminaries

^are

required.

It is useful to

work with the

rescaling

^{of u~}

given by

^{Vé, as defined at} the end of the last section. Given R >

0,

^we denote

by

^the

Aé) R~,

a similar definition has the set The next lemma states that Vc: is small in

H1-norm

away from the ^set ~ ⁼

LEMMA 2.2. - There exists a C > 0 such

that, given

^{R >}

0,

^one has

for

^{all E}

sufficiently

^small.

Proof -

^{Given R > >}

^0,

^we may choose smooth cut-off functions

0 ~

^{1 such that}

and

]

^{Then set}

r~~

^{= 1 -}

Using

^{the test function}

in

(1.24), equation

^satisfied

^weakly by

~F, ^one gets

where w= is

given by ( 1.21 ).

Observe that _w= is

uniformly bounded, by

^a

bound

possibly depending

^{on ~~I.}

Using this,

the choice of the fact that

Annales de l’Institut Henri Poincaré - Analyse ^{non linéaire}

VE is

uniformly

bounded in and the definition of

j,

the desired estimate

(2.2)

^follows

immediately

^from

(2.4).

^D.

A second

preliminary

^{result we will need is}

given

^{in the}

following

lemma

inspired by

the work in

[4].

Proof -

^We

begin by showing

^{that uJ a on}

{:rl

⁼

0~.

^Standard

regularity

arguments

yield

that v is in

C1 ( f~ ~ )

ⁿ

~I2 ( f~’~’ ) . ^Moreover,

~c ~ ( :z~ ) . Wu ( ~; )

^{-~ 0}

as I

^{-~ ~c .}

Using ~-

^{as a test} function in

equation (2.5),

^{we obtain}

But the first summand in the above

quantity

^{is zero, while}

F (.5 ) > F ( s ),

with

equality only

^{if s a.}

^Thus, v(o, ~t;’)

^a.

Finally,

^to prove that a for x~ 1 >

0, we just

consider the test function

cP == (v -

ⁱⁿ

equation (2.5),

^{and the conclusion}

0

readily

^{follows. D}

Proof of

Lemma 2.1. - We will base the

proof

^{in an indirect}

argument.

Thus,

^assume that

(2.1)

^{does not}

hold, namely

^{that for some} sequence

~~ ^{- 0 we have}

We will see that

(2.7)

^is

impossible provided

^{that AI was chosen}

sufficiently large.

^In fact this is a consequence of ^the

following ^claim,

^{main step in}

the

proof:

^if

(2.7) holds,

^then

Vol. 15, n’ 2-1998.

We will show

(2.8).

^{We start}

proving

^{that the}

sequence

concentrates

somewhere near more

precisely,

^we show that there exist numbers S >

0,

p ^> 0 such that

To see

this,

^we first observe that from

(2.7),

^{there is a A > 0 such that}

then,

^{Lemma 2.2}

imply

that for all R > 0

large enough

Now assume that

(2.9)

is false. Then we may ^assume that for all S > 0

we have

for all S > 0.

Moreover,

^{is a} bounded sequence in Then

applying

the concentration compactness

principle (see

Lemma 1.1 in

[7]

^or

Lemma 2.18 in

[2]),

^we obtain that

for each R >

0, then,

ⁱⁿ

particular

where s is as in

(f2). Using vR

^{as a test function in}

^(1.25),

^the

^equation

satisfied ^we get

Annales de l’Institut Henri Poincaré - Analyse ^{non linéaire}

Hence,

choosing R and j large enough,

^{we obtain from the above}

^estimate,

an immediate contradiction with

(2.10).

This shows the

validity

^of

(2.9).

Thus,

^we may ^assume there is a sequence ?/j ^G with

Let us now write ⁼ +

y).

^Since v~ is ^a bounded sequence in

we may ^assume it converges

weakly

^{to a v E}

H~ (l~=~~).

^Assume

first that

Set ~ ^{== G}

A-~

^{and assume that} _;r~ ^{2014~ .c G}

A~.

^Since f~ satisfies in

{-~}

⁺

A~

^the

^equation

it follows that v satisfies in

where b ⁼

V(x). Moreover, v t. 0,

^{thanks to}

(2.13).

^{On the other}

hand,

if C we will have that v satisfies ^an

equation

^of

the form

(2.5),

^{so that Lemma 2.3}

implies

that v satisfies

(2.15).

^{In both}

cases v is the

unique

^critical

point

^{of the} functional

I~

defined in

(1.7),

with b

sup{V(x) x

^G

1~ i ~ bi

^{+ 8. Then we have}

On the other

hand, elliptic regularity implies,

ⁱⁿ

particular,

^that vv~

converges

strongly

^{in the}

^H1-sense

^over compacts.

Passing

^{to a further}

subsequence

^if necessary, ^we may find a sequence of

positive

^numbers

-I- ~c such that

Thus,

combining (2.7)

^and

(2.16)

^{we find} that there

exists r]

^{> 0 such}

that for all

large j

Vol. 15. n ’ 2-1998.

Using (2.17)

^and

applying

^a

concentration-compactness

argument, ^similar

to that we used above, to the sequence obtained after

multiplying ^by

a suitable cut-off function

vanishing

^on

(~~~ ),

^we will end up with the existence of an S > 0 and a

sequence

^E

11.1-’ B

^{such that}

Thus, we have

again,

^after

passing

^{to a}

subsequence,

the weak convergence of

( -

⁺

~~ )

^{to a nonzero E}

Moreover, w

satisfies the

equation

where b = V() ,

^{with E}

^{11.i .}

^From

(2.18) ~ ^0,

^hence

I() ~

^{ct .}

Our next claim is that

To

verify ^this,

^{we recall satisfies on the}

equation

We use in this

equation

^{a test} function of the form

where 03C8

^{is a function = 0 for s 1 and = 1 for}

s > 2.

Denoting

^{= U the} conclusion we

obtain,

after a direct estimate is that

Then,

it follows that

Annales de l’Institut Henri Poincaré - Analyse ^{non linéaire}

so that

from where

(2.19)

^follows since R is

arbitrary,

and the claim

(2.8)

^is

thus checked.

We observe that a similar argument

applies

^to the functional

J~,

^{so we}

also have

The definition of the total functional

E~

ⁱⁿ

(1.10)

^and

(1.11),

^thus

yields

But, using

the upper estimate in the critical value

C~~

⁼

E~~ (~c~? ) ^provided

by

^Lemma

^1.2,

^{and the}

inequality ^above,

^{we obtain}

Therefore,

^{if M was such that}

we obtain that our

original assumption ^(2.7)

^was

impossible.

^{This concludes}

the

proof

^{of Lemma 2.1. D}

We assume in the rest of this section that M was chosen so

large

^that

Lemma 2.1 I holds true. Our next lemma is LEMMA 2.4

Proof -

^We

begin by showing

Yo!. 15. n ~ 2-1998.

Notice that it suffices to show that

In

fact,

concentration arguments ^{similar to those}

employed

^{in the}

proof

^of

Lemma 2.1

give

^{that then}

where b ⁼ with ~z; E

Then assume that

(2.23)

^{does not hold for}

some i,

say ^{i = 1.} Then,

along

^a sequence ~~ ^{~ 0 we} get

But Lemma 2.1 I

implies ^that,

^{for all i. =}

1; ... , _K

The definition of

E~, together

^with

(2.25), (2.26),

^{and Lemma 2.2 then}

yields

that is

certainly impossible by

the choice of the _~.t made in

(1.8).

^This

concludes the

proof

^of

(2.22).

Next we show that

(2.21)

holds. Assume that for Ii ⁼ 1 we have

along

a sequence that

Then we will get, ^{from estimate}

(2.22)

^{and Lemma}

2.1,

which is

impossible.

This concludes the

proof

^{of Lemma 2.4. 0}

Conclusion

of

^the

Proof of

Theorem o.1. - To

begin ^with,

^{we show that}

In

fact,

^otherwise there would exist a sequence ~~ ^{-~ 0 and E}

~11~

^with

> b > 0.

Assuming

^{-~ ~z~ E}

c~,11,

^and

using

^{Lemma 2.3, v~-e}

Annales de l’Institut Henri Poincaré - Analyse ^{non linéaire}

obtain that the sequence ⁺ converges

weakly

ⁱⁿ

^H~

to a nonzero solution 2J of

Since b ⁼

Z~(~ )

^>

^bi,

^{we have that}

I~ (v)

^{> c,. But this}

imply

^that

which is

impossible

in view of Lemma 2.4. Hence

(2.29)

holds. Note that

(2.29) implies,

ⁱⁿ

particular,

^that ~c~. a on for all i ⁼

1... K,

^if

ê is

sufficiently

^small.

Thus the

function ~) = (uc -

^{is in}

Ho (SZ),

^{and we can use it}

as a test function in

(1.22),

^the

equation

^satisfied

by

^{IUC’ We}

immediately

conclude that

actually 03C6 ~ 0,

^{in other}

words,

^u

~ a on fl B

^{A. The}

conclusion is that _~u~ is

actually

^{a solution to the}

original equation (0.6)

^for

all small _c, one of the desired features of

Finally,

the fact that ⁼ _cl, > 0

implies

^that

concentration must occur around some x G

A.;.

^{We must then have}

Ij(~z; )

⁼

^bi,

for otherwise ^a contradiction similar ^to

(2.28)

would arise.

The concentration fact

implies

^the presence of at least one local maximum in each

A;,

^{so that} ~u~ is a

K-peak

^{solution. The}

uniqueness

^{of these}

^maxima,

as well as the

remaining decay

assertions of the theorem follow

similarly

to their

analogues

^{shown in}

[3]

^{for the}

one-peak

^case. This concludes the

proof

^{of the theorem. D}

Remark 2.1. - If instead of

hypothesis (f4)

^{we assume}

(f4’),

^{as indicated}

in the

introduction,

^the

proof

of Theorem 0.1 ^can be carried ^out in a similar way, after

choosing

a-1 i so small that no critical value of the functional

IV,

other than

c(b),

^{exist in}

(0,

^{ct +}

3~1 ),

for all b E

(bl . bl

⁺

b),

^{~i, = 1... K.}

We

modify

^the

penalization

^as

and choose M so that _i is

large enough.

3. APPENDIX

We devote the

Appendix

^{to the}

proof

^{of Lemma 1.3.} Since many steps in the

proof

^use arguments

already given

in Section 2, ^{at certain}

points

we don’t

give

^{all details.}

Voi. i5. n 2-1998.

Pr oof of

Lemma 1.3. - An upper estimate of the form

follows from the use of a test

path

constructed ^as in the

beginning

^{of the}

proof

^{of Lemma 1.2.}

On the other

hand,

it is standard to check that the functional

~T~

^satisfies

P.S. in

H~ (11.z ),

^so that the Mountain Pass Theorem

implies

^that

d~

^{is a}

critical value for it. Let w~ be an associated critical

point.

^{Then w~ is a}

nonzero solution of the

equation

We

begin by observing

that the definition of

g(x, u)

and the Maximum

Principle

^{makes it}

impossible

^{that w‘ attains a} local maximum somewhere in

Ai.

^{Thus let} x~ E

Ai

^{be a} maximum of w~. Then ^we have

w~ > b

^>

0, uniformly

^{in E. We} consider the

rescaling

^of w~

given by v~ (~)

⁼

~ ~~),

^{and take a} sequence ~~ ^~ 0.

Then,

for a

subsequence

of ~~ which ^we relabel in the same way, ^we have that ~ :.~,~ G

A,.

Moreover,

^estimate

(3.1)

and the fact that w~ solves

equation (3.2) imply

1 or

equivalently, ~v~~H1 ~

C. This fact and

elliptic

estimates allow us to assume, without loss of

generality,

^that

_v~~

J --~ ?;

weakly

ⁱⁿ

^H1

^and

locally strongly

^{in the}

C1-sense,

where v is a nontrivial solution of an

equation

of the form

(2.5),

^so that Lemma 2.3 is

applicable

to conclude that v satisfies

Set b ⁼

y’ ( ~z; ) .

^{Then b >}

b.i and, using

arguments ^{as in the}

proof

^{of Lemma}

2.1,

^{we must have}

where

It’

was defined in

(1.7).

But b ⁼

~T (:~ )

^>

^{b; ,}

^{so we have that}

IL (~)

^{> Since we} have established that every sequence ^{--~ 0 has a}

subsequence

^{such that}

(3.3)

^{holds, we}

then

conclude di~

^>

(c,

^{+ as} desired. This finishes the

proof.

^D

Annales cle l’Institut Henri Poincaré - Analyse ^{non linéaire}

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799~;

/~, 7996J

Vol. 15, n ’ 2-1998.